# Dynamic programming on graphs with bounded treewidth

## Abstract

In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth ≤*k*, (or equivalently, the class of partial *k*-trees), for fixed *k*. We introduce two classes of graph decision problems, LCC and ECC, and subclasses *C*-LCC, and *C*-ECC. We show that each problem in LCC (or *C*-LCC) is solvable in polynomial (*O*(*n*^{ C })) time, when restricted to graphs with fixed upper bounds on the treewidth and degree; and that each problem in ECC (or *C*-ECC) is solvable in polynomial (*O*(*n*^{ C })) time, when restricted to graphs with a fixed upper bound on the treewidth (with given corresponding tree-decomposition). Also, problems in *C*-LCC and *C*-ECC are solvable in polynomial time for graphs with a logarithmic treewidth, and in the case of *C*-LCC-problems, a fixed upper bound on the degree of the graph.

Also, we show for a large number of graph decision problems, their membership in LCC, ECC, *C*-LCC and/or *C*-ECC, thus showing the existence of *O*(*n*^{ C }) or polynomial algorithms for these problems, restricted to the graphs with bounded treewidth (and bounded degree). In several cases, *C*=1, hence our method gives in these cases linear algorithms.

For several NP-complete problems, and subclasses of the graphs with bounded treewidth, polynomial algorithms have been obtained. In a certain sense, the results in this paper unify these results.

## Keywords

Treewidth partial*k*-trees graph decision problems restrictions of NP-complete problems polynomial time algorithms dynamic programming local condition compositions

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